A mathematical model for the burden of diabetes and its complications
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Abstract
Background
The incidence and prevalence of diabetes are increasing all over the world. Complications of diabetes constitute a burden for the individuals and the whole society.
Methods
In the present paper, ordinary differential equations and numerical approximations are used to monitor the size of populations of diabetes with and without complications.
Results
Different scenarios are discussed according to a set of parameters and the dynamical evolution of the population from the stage of diabetes to the stage of diabetes with complications is clearly illustrated.
Conclusions
The model shows how efficient and costeffective strategies can be obtained by acting on diabetes incidence and/or controlling the evolution to the stage of complications.
Keywords
Kidney Failure Euler Method Diabetic Population International Dollar Health Decision MakerBackground
It is now commonly admitted that diabetes is sweeping the globe as a silent epidemic largely contributing to the growing burden of noncommunicable diseases and mainly encouraged by decreasing levels of activity and increasing prevalence of obesity. The recent reports released by the World Health Organization [1] and the International Diabetes Federation [2] are alarming. In 2003, it was estimated that 194 million people were diabetic, representing a global prevalence exceeding 3% (5.1% for those aged 20 to 79) of the world population. The trend is increasing and the number is expected to reach 333 million (6.3%) by the year 2025. Moreover, for the first time, an estimation of 314 million (8.2%) is given for people in the prediabetic stage which constitutes a compartment from which at least one third will evolve to the diabetic stage after 10 years.
Estimated numbers of diabetics (in Million)
Country  2003  2025 

India  35.5  73.5 
China  23.8  46.1 
United States  16  23.1 
Russia  9.7  10.7 
Japan  6.7  7.1 
Germany  6.3  7.1 
Pakistan  6.2  11.6 
Brazil  5.7  10.7 
Mexico  4.4  9 
Egypt  3.9  7.8 

the leading cause of endstage kidney failure necessitating dialysis or transplantation,

the leading cause of blindness in people of working age,

the leading cause of amputation,

the first cause with other risk factors of mortality and morbidity by cardiovascular diseases.
The burden of diabetes and its complications
 1.
Cost directly related to the diagnosis and management of diabetes without complications. This includes the inpatient and outpatient care, means of treatment by insulin or tablets and the equipment of self control (blood and urine testing).
 2.
Costs generated by complications of diabetes. These are difficult to quantify because diabetes is linked to micro and macro vascular diseases such as heart disease, kidney failure, eye disease and amputation. Moreover, diabetes may add a cost of care by complicating other unrelated medical situations like infections, accidents and surgery.
 3.
Indirect costs correlated to the quality of life and the economic productivity which can be somehow estimated by the degree of disability.
In order to facilitate meaningful comparisons across world regions, costs are often expressed in international dollars (an international dollar has the same purchasing power as one US dollar has in the USA) and costeffectiveness is measured in terms of years lived with disability (YLD) or disability adjusted life years (DALY) [11, 12]. Studies in different countries have shown that diabetes is a costly disease accounting for between 2.5 and 15% of the total healthcare expenditure. For the age category 20–79, the world annual direct cost is estimated to be over 153 billion and expected to double in 2025 [2, 13, 14, 15, 16]. According to the National Institute of Diabetes and Digestive Kidney Disease (NIDDK) and the American Diabetes Association, diabetes was the sixth leading cause of death in 1999 with a direct cost of $44 billion and an indirect cost of $54 billion annually. In 2002, the direct and indirect cost totaled $132 billion [14]. In France, an estimation of $5.7 billion was given for the direct cost of diabetes [5], whereas, an equivalent cost of £5.2 billion, representing approximately 9% of the annual national health service (NHS) budget, was given for UK in 2000 [15]. The burden affects also developing countries as stressed by the different authors who attended the seventh congress of the PanAfrican diabetes study group in 2001 [16] and the Metabolic Syndrome type II Diabetes and Artherosclerosis Congress in 2004 [17]. In these, countries, until recently, it was widely believed that economic development was a necessary prerequisite for improving a population health status and the health was often classified as a non productive sector. Now, politicians and health policy makers are timidly recognizing that investing in people's health is a necessary condition for economic development but energetic decisions are needed for the adoption of urgent and consequent strategies. The need for such strategies is enhanced by the fact that risk factors like cholesterol, tobacco, blood pressure, and obesity are no more a specificity of industrialized countries, they are becoming more prevalent in developing nations, where they double the burden of infectious diseases that have always afflicted poorer countries [10].
The literature dealing with modeling for diabetes is mainly concerned with glucose and insulin dynamics [6, 18, 19, 20], the epidemiology of the disease [21, 22, 23] and economic cost and risk models [24, 25, 26, 27, 28, 29]. In previous papers, the authors considered continuous and matrix models for age structured populations of diabetics [30, 31] and Dynamics of a disabled population in Morocco [32]. In the present paper, while stressing the growing burden of disease caused by diabetes and its complications, a model is proposed to monitor the size of the diabetic population and to deal with the evolution from the stage of diabetes without complications to the stage of diabetes with complications. Parameters can be handled to illustrate the effect of an increasing or decreasing incidence of diabetes and its complications. Consequently, different strategies can be adopted. The main purpose is to show that investment in primary health care is a necessary and costeffective strategy that allow to control the incidences of diabetes and its complications and hence, to convince policy makers that bold decisions must be taken for a sustainable development which ensures better quality of life and wellbeing for the present and future generations of humans.
Methods
The mathematical model
Suppose that C = C(t) and D = D(t) represent the numbers of diabetics with and without complications, respectively, and let N = N(t) = C(t) + D(t) denote the size of the population of diabetics at time t (see Nomenclature). Then, as was noted earlier, N(t) ~ 3% of the world population. Let I = I(t) denote the incidence of diabetes mellitus. The model parameters to be incorporated are μ (the natural mortality rate), λ (the probability of a diabetic person developing a complication), γ (the rate at which complications are cured), ν (the rate at which diabetic patients with complications become severely disabled) and δ (the mortality rate due to complications).
The diagram shows that I = I(t) cases are diagnosed in a time interval of length t and are assumed to have no complications upon diagnosis. In that same time interval, the number of sufferers without complications, D = D(t), is seen to decrease by the amounts μD (natural mortality) and λD (sufferers who develop complications), and to increase by the amount γC (sufferers whose complications are cured). During this time interval, the number of diabetics with complications is increased by the aforementioned amount γC and by the amount μC (natural mortality), νC (patients who become severely disabled and whose disabilities cannot be cured) and δC (those who die from their complications).
These rates of change are formalized by the ordinary differential equations (ODEs)
which, since N(t) = D(t) + C(t), give rise to the initialvalue problem (IVP)
C'(t) = (λ + θ)C(t) + λN(t), t > 0; C(0) = C_{0} (1)
N'(t) = I(t)  (ν + δ)C(t)  μN(t), t > 0; N(0) = N_{0} (2)
where θ = γ + μ + ν + δ, Open image in new window and C_{0}, N_{0} are the initial values of C(t) and N(t), respectively.
In the case when the probability of a diabetic person developing a complication, λ, is constant, the model equations (1), (2) are linear in C(t) and N(t): this linear model will be discussed in the following paragraph. The nonlinear model corresponding to a variable λ will be considered by the authors in another paper more devoted to numerical analysis.
The linear case
The critical point and its stability property
The probability of developing a complication, λ, will be estimated to have the constant value
The initialvalue problem (1), (2) may consequently be written in matrixvector form as
x'(t) = Ax(t) + b(t), t > 0;x(0) = X_{0} (4)
in which
Suppose that I is the steadystate value of the incidence, then the model reaches its critical point when dC/dt and dN/dt given in (1) and (2) vanish simultaneously, that is when
λN  (λ + θ)C = 0, (6)
I  μN  (ν + δ)C = 0. (7)
Solving (6) and (7)gives
and
The eigenvalues of the matrix A, χ_{1}, χ_{2}, are the roots of the quadratic equation (the characteristic equation)
χ^{2} + (λ + θ + μ)χ + μ(λ + θ) + λ(ν + δ) = 0. (9)
The discriminant, Δ, of this equation is given by
Δ = (λ + θ + μ)^{2}  4 [μ(λ + θ) + λ(ν + δ)]
and, recalling that θ = ν + μ + δ + γ it follows that
Δ <λ + μ + δ + γ + 2ν.
Solving (9) gives
 (a)
Δ > 0, χ_{1}, χ_{2} are both real and negative;
 (b)
Δ = 0, χ_{1} = χ_{2} are real and negative;
 (c)
Δ < 0, χ_{1} and χ_{2} are complex conjugate with negative real parts.
it may be concluded, therefore, that the critical point (C*, N*) of (1), (2), given by (8), is stable.
Numerical solution and stability
It may be shown that the solution x(t) of the IVP (4) satisfies the recurrence relation
where l > 0 is an increment in t (the time step). This recurrence relation may be used to generate x(t_{n + 1}) in terms of x(t_{ n }), thus monitoring C(t) and N(t) at the discrete points t = t_{ n }= nl(n = 0, 1, 2, ...).
One very simple way of estimating x(t + l) is to approximate to second order the integral in (11) by the trapezoidal rule, viz.
and then to replace, also to second order, exp(lA) in (11) and (12) by its (1,1) Padé approximant
exp(lA) = (E  1/2lA)^{1} (E + 1/2lA) (13)
where E is the identity matrix of order two.
Denoting by X_{ n }the numerical approximation to x(t_{ n }) calculated using (11), (13), it may be shown, by substituting (12) with (13) in (11) and then by premultiplying by (E  1/2lA), that
(E  1/2lA)X_{n + 1}= (E + 1/2lA)X_{ n } (14)
+
1/2l[(E  1/2lA)b_{n + 1}+ (E + 1/2lA)b_{ n }];n = 0, 1, 2, ...
where X_{ n }= (C_{ n }, N_{ n })^{T}, T denoting transpose, and b_{ n }= (0, I_{ n })^{T} with I_{ n }= I(t_{ n }). It may then be shown that C_{n + 1}and N_{n + 1}(n = 0, 1, 2,...) may be determined by solving the algebraic equations given by
(Method 1) (1 + 1/2l(λ + θ))C_{n + 1} 1/2lλN_{n + 1}=
[1  1/2l(λ + θ)]C_{ n }+ 1/2lλN_{ n } 1/4l^{2}λ(I_{n + 1} I_{ n }) (15)
and
1/2l(ν + δ)C_{n + 1}+ (1 + 1/2lμ)N_{n + 1}=
1/2l(ν + δ)C_{ n }+ (1  1/2lμ)N_{ n }
+1/2l(1 + lμ)I_{n + 1}+ 1/2l(1  lμ)I_{ n } (16)
assuming convergence, C_{n + 1}= C_{ n }= C, N_{n + 1}= N_{ n }= N and I_{n + 1}= I_{ n }= I, say, then equations (15) and (16) become
(λ + θ)C  λN = 0, (17)
(ν + δ)C + μN = I, (18)
respectively. Obviously (17) and (18) are the same as (6) and (7) and so the fixed point (C^{+}, N^{+}) of the numerical solution sequence (C_{ n }, N_{ n }), n = 0, 1, 2,... is the same as the critical point (C*, N*) of the linear initial valueproblem.
For comparison purpose, the IVP (1), (2) was also solved using the wellknown Euler method (a firstorder method) given by
(Method 2) C_{n + 1}= [1  l(λ + θ)]C_{ n }+ lλN_{ n } (19)
N_{n + 1}= l(ν + δ)C_{ n }+ (1  lμ) + lIN_{ n } (20)
The method1 is unconditionally stable whereas the Euler method is conditionally stable provided ([33])
Numerical experiments
Taking I(t) = I to be constant equations (15) and (16) simplify to
(1 + 1/2l(λ + θ))C_{n + 1} 1/2lλN_{n + 1}=
[1  1/2l(λ + θ)]C_{ n }+ 1/2lλN_{ n } (22)

1/2l(ν + δ)C_{ n }+ (1  1/2lμ)N_{ n }+ lI, (23)
Parameter vlaues used in numerical experiments
Parameter  Value yr^{1} 

ν  0.05 
δ  0.05 
μ  0.02 
γ  0.08 or 0 
C* = 47000000 and N* = 61100000. (24)
Using Matlab, four numerical experiments were carried out taking as initial conditions
C_{0} = C* ± 500 and N_{0} = N* ± 500. (25)
Fixed point values (× 1O^{7}) for the linear model using different values of l, I = 6 × 10^{7}, λ = 0.66
N ^{+}  N ^{+}  C ^{+}  C ^{+}  

l yr  Euler  Method1  Euler  Method1 
0.01  6.08  6.08  4.75  4.75 
0.02  6.11  6.11  4.77  4.77 
0.05  6.11  6.11  4.7  4.7 
0.1  6.11  6.11  4.7  4.7 
0.2  6.11  6.11  4.7  4.7 
0.5  6.11  6.11  4.7  4.7 
1  6.11  6.11  4.7  4.7 
2  6.11  6.11  4.7  4.7 
2.5  div  6.11  div  4.7 
3  div  6.11  div  4.7 
3.5  div  6.11  div  4.7 
4  div  6.11  div  4.7 
The initial conditions in (25) are close to the steadystate solutions C* and N*. Other initial conditions, further from C* and N*, will converge to the same values of C^{+} and N^{+} for the same value of l, though these values will be reached at different times.
Retaining the parameters values shown in Table 2 the effect of the choice of time step was monitored in a series of 11 further experiments. The fixedpoint values, C^{+} and N^{+} to which convergence occurred are shown in Table 3, where it may be seen that, for the larger values of l (l ≥ 0.5yr), there is very close agreement with the criticalpoint values given in (24).
For l ≤ 2yr (approximately), the two methods give similar results but for l > 2.5yr (approximately (21)) the Euler method diverged. The values of C^{+} and N^{+} using Euler method are given in Table 3.
It may be concluded from these results that the Euler method may be used with confidence if the diabetic population is to be monitored at time intervals up to approximately two years using the linear model (1), (2). However, to monitor the population less frequently the numerical method (Method 1) should be used.
Results and discussion
Output of the nine scenarios according to each level of incidence I and complications λ (× 10^{7})
I λ  L  M  H 

L  C = 0.72  C = 2.18  C = 4.36 
N = 1.36  N = 4.09  N = 8.18  
M  C = 0.75  C = 0.22  C = 4.55 
N = 1.20  N = 3.6  N = 7.21  
H  C = 0.79  C = 2.38  C = 4.77 
N = 1.0  N = 3.05  N = 6.11 

Reduce new blindness due to diabetes by one third or more

Reduce numbers of people entering endstage diabetic renal failure by at least one third

Reduce by one half the rate of limb amputations for diabetic gangrene

Cut morbidity and mortality from coronary disease in the diabetic by vigorous programmes of risk factor reduction

Achieve pregnancy outcome in the diabetic woman that approximates that of the nondiabetic woman.
However, although most of developed countries have reacted by pragmatic measures, the trend remain globally passive mainly because developing countries have been, so far, satisfied with adopting national conventions and adhering to international recommendations instead of working in the field. As stressed earlier, this behaviour can be partly explained by lack of means and poor budget affected to health care but, in general, bad management and absence of goodwill assume a large part of responsibility. The illustration yielded by our mathematical model confirms the diagnosis and the recommendations given by specialists and experts in the field of diabetes and health management in general. Moreover, it gives to health decision makers guide lines of comparison between the social and economic costs of uncontrolled diabetes, and the benefit gained by a productive investment in primary healthcare.
Conclusion
In this paper, a mathematical model was proposed to deal with the dynamics of a population of diabetes. The model was formalized by a system of ordinary differential equations, then numerical approximations were used to obtain numerical results. Although linear and nonlinear cases were considered, for sake of clarity and simplicity, only numerical results of the linear model were given.
The model showed clearly the results given according to different scenarios. The main purpose was to stress the importance to control the incidence of diabetes and its complications and hence to convince decision makers that investment in healthcare is a costeffective strategy.
Authors contributions
AB participated to the proposition and discussion of the model.
EHT elaborated numerical analysis and English writing.
KA elaborated numerical experiments.
AC elaborated numerical procedures and T_{E}X writing.
Appendix
t : time,
l : increment in t (the time step),
C(t) : number of diabetics with complications,
D(t) : number of diabetics without complications,
N(t) : number of diabetics (N = C + D),
I(t) : incidence of diabetes mellitus,
J : jacobien,
μ : natural mortality rate,
λ : probability of developing a complication,
γ : rate at which complications are cured,
ν: rate at which patients with complications become severely disabled,
δ : mortality rate due to complications,
θ = μ + δ + γ + ν,
χ_{1}, χ_{2} : eigenvalues,
C_{0}, N_{0} : initial values of C and N,
C*, N* : criticalpoint values of C and N,
C^{+}, N^{+} : fixedpoint values of C and N,
x: x = [C, N]^{T}, T denoting transpose,
X_{0} : X_{0} = [C_{0}, N_{0}]^{T},
A : constant square matrix of order two (linear model),
b(t): b(t) = [0, I(t)]^{T},
C_{ n }, N_{ n }: approximations to C(nl), N(nl),
X_{ n }: X_{ n }= [C_{ n }, N_{ n }]^{T},
I_{ n }: I_{ n }= I(nl).
Notes
Acknowledgements
One of the authors (A.B) is grateful to the British Council (Morocco) for financial support during the period of research. This paper is dedicated to Wiam Boutayeb and David M. Barlett.
Supplementary material
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